LMFDB - Embedded newform 2450.2.a.bt.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} + 25 $ x^4 - 12*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.05231$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.05231 q^{3} +1.00000 q^{4} -3.05231 q^{6} -1.00000 q^{8} +6.31662 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.05231 q^{3} +1.00000 q^{4} -3.05231 q^{6} -1.00000 q^{8} +6.31662 q^{9} +5.31662 q^{11} +3.05231 q^{12} +3.27620 q^{13} +1.00000 q^{16} -7.29496 q^{17} -6.31662 q^{18} +1.63810 q^{19} -5.31662 q^{22} +8.63325 q^{23} -3.05231 q^{24} -3.27620 q^{26} +10.1234 q^{27} -2.63325 q^{29} -5.65685 q^{31} -1.00000 q^{32} +16.2280 q^{33} +7.29496 q^{34} +6.31662 q^{36} +4.63325 q^{37} -1.63810 q^{38} +10.0000 q^{39} +2.60454 q^{41} -6.63325 q^{43} +5.31662 q^{44} -8.63325 q^{46} -6.10463 q^{47} +3.05231 q^{48} -22.2665 q^{51} +3.27620 q^{52} -8.00000 q^{53} -10.1234 q^{54} +5.00000 q^{57} +2.63325 q^{58} +1.86199 q^{59} +6.10463 q^{61} +5.65685 q^{62} +1.00000 q^{64} -16.2280 q^{66} -1.63325 q^{67} -7.29496 q^{68} +26.3514 q^{69} -2.00000 q^{71} -6.31662 q^{72} -7.29496 q^{73} -4.63325 q^{74} +1.63810 q^{76} -10.0000 q^{78} +1.36675 q^{79} +11.9499 q^{81} -2.60454 q^{82} -8.26139 q^{83} +6.63325 q^{86} -8.03751 q^{87} -5.31662 q^{88} +4.91430 q^{89} +8.63325 q^{92} -17.2665 q^{93} +6.10463 q^{94} -3.05231 q^{96} +4.24264 q^{97} +33.5831 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{16} - 12 q^{18} - 8 q^{22} + 8 q^{23} + 16 q^{29} - 4 q^{32} + 12 q^{36} - 8 q^{37} + 40 q^{39} + 8 q^{44} - 8 q^{46} - 36 q^{51} - 32 q^{53} + 20 q^{57} - 16 q^{58} + 4 q^{64} + 20 q^{67} - 8 q^{71} - 12 q^{72} + 8 q^{74} - 40 q^{78} + 32 q^{79} + 8 q^{81} - 8 q^{88} + 8 q^{92} - 16 q^{93} + 68 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^16 - 12 * q^18 - 8 * q^22 + 8 * q^23 + 16 * q^29 - 4 * q^32 + 12 * q^36 - 8 * q^37 + 40 * q^39 + 8 * q^44 - 8 * q^46 - 36 * q^51 - 32 * q^53 + 20 * q^57 - 16 * q^58 + 4 * q^64 + 20 * q^67 - 8 * q^71 - 12 * q^72 + 8 * q^74 - 40 * q^78 + 32 * q^79 + 8 * q^81 - 8 * q^88 + 8 * q^92 - 16 * q^93 + 68 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.05231	1.76225	0.881127	−	0.472879i	$-0.156785\pi$
$3$	3.05231	1.76225	0.881127	+	0.472879i	$0.156785\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.05231	−1.24610
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	6.31662	2.10554
$10$	0	0
$11$	5.31662	1.60302	0.801511	−	0.597980i	$-0.204030\pi$
$11$	5.31662	1.60302	0.801511	+	0.597980i	$0.204030\pi$
$12$	3.05231	0.881127
$13$	3.27620	0.908655	0.454328	−	0.890835i	$-0.349880\pi$
$13$	3.27620	0.908655	0.454328	+	0.890835i	$0.349880\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.29496	−1.76929	−0.884643	−	0.466268i	$-0.845598\pi$
$17$	−7.29496	−1.76929	−0.884643	+	0.466268i	$0.845598\pi$
$18$	−6.31662	−1.48884
$19$	1.63810	0.375806	0.187903	−	0.982188i	$-0.439831\pi$
$19$	1.63810	0.375806	0.187903	+	0.982188i	$0.439831\pi$
$20$	0	0
$21$	0	0
$22$	−5.31662	−1.13351
$23$	8.63325	1.80016	0.900078	−	0.435728i	$-0.143509\pi$
$23$	8.63325	1.80016	0.900078	+	0.435728i	$0.143509\pi$
$24$	−3.05231	−0.623051
$25$	0	0
$26$	−3.27620	−0.642516
$27$	10.1234	1.94825
$28$	0	0
$29$	−2.63325	−0.488982	−0.244491	−	0.969652i	$-0.578621\pi$
$29$	−2.63325	−0.488982	−0.244491	+	0.969652i	$0.578621\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	−1.00000	−0.176777
$33$	16.2280	2.82493
$34$	7.29496	1.25107
$35$	0	0
$36$	6.31662	1.05277
$37$	4.63325	0.761702	0.380851	−	0.924637i	$-0.375631\pi$
$37$	4.63325	0.761702	0.380851	+	0.924637i	$0.375631\pi$
$38$	−1.63810	−0.265735
$39$	10.0000	1.60128
$40$	0	0
$41$	2.60454	0.406761	0.203380	−	0.979100i	$-0.434807\pi$
$41$	2.60454	0.406761	0.203380	+	0.979100i	$0.434807\pi$
$42$	0	0
$43$	−6.63325	−1.01156	−0.505781	−	0.862662i	$-0.668795\pi$
$43$	−6.63325	−1.01156	−0.505781	+	0.862662i	$0.668795\pi$
$44$	5.31662	0.801511
$45$	0	0
$46$	−8.63325	−1.27290
$47$	−6.10463	−0.890452	−0.445226	−	0.895418i	$-0.646877\pi$
$47$	−6.10463	−0.890452	−0.445226	+	0.895418i	$0.646877\pi$
$48$	3.05231	0.440564
$49$	0	0
$50$	0	0
$51$	−22.2665	−3.11793
$52$	3.27620	0.454328
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	−10.1234	−1.37762
$55$	0	0
$56$	0	0
$57$	5.00000	0.662266
$58$	2.63325	0.345763
$59$	1.86199	0.242410	0.121205	−	0.992627i	$-0.461324\pi$
$59$	1.86199	0.242410	0.121205	+	0.992627i	$0.461324\pi$
$60$	0	0
$61$	6.10463	0.781618	0.390809	−	0.920472i	$-0.372195\pi$
$61$	6.10463	0.781618	0.390809	+	0.920472i	$0.372195\pi$
$62$	5.65685	0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−16.2280	−1.99753
$67$	−1.63325	−0.199533	−0.0997666	−	0.995011i	$-0.531810\pi$
$67$	−1.63325	−0.199533	−0.0997666	+	0.995011i	$0.531810\pi$
$68$	−7.29496	−0.884643
$69$	26.3514	3.17234
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−6.31662	−0.744421
$73$	−7.29496	−0.853810	−0.426905	−	0.904296i	$-0.640396\pi$
$73$	−7.29496	−0.853810	−0.426905	+	0.904296i	$0.640396\pi$
$74$	−4.63325	−0.538604
$75$	0	0
$76$	1.63810	0.187903
$77$	0	0
$78$	−10.0000	−1.13228
$79$	1.36675	0.153771	0.0768857	−	0.997040i	$-0.475502\pi$
$79$	1.36675	0.153771	0.0768857	+	0.997040i	$0.475502\pi$
$80$	0	0
$81$	11.9499	1.32776
$82$	−2.60454	−0.287623
$83$	−8.26139	−0.906806	−0.453403	−	0.891306i	$-0.649790\pi$
$83$	−8.26139	−0.906806	−0.453403	+	0.891306i	$0.649790\pi$
$84$	0	0
$85$	0	0
$86$	6.63325	0.715282
$87$	−8.03751	−0.861711
$88$	−5.31662	−0.566754
$89$	4.91430	0.520915	0.260458	−	0.965485i	$-0.416127\pi$
$89$	4.91430	0.520915	0.260458	+	0.965485i	$0.416127\pi$
$90$	0	0
$91$	0	0
$92$	8.63325	0.900078
$93$	−17.2665	−1.79045
$94$	6.10463	0.629644
$95$	0	0
$96$	−3.05231	−0.311526
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	33.5831	3.37523
$100$	0	0
$101$	15.0377	1.49631	0.748153	−	0.663526i	$-0.230941\pi$
$101$	15.0377	1.49631	0.748153	+	0.663526i	$0.230941\pi$
$102$	22.2665	2.20471
$103$	11.3137	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$103$	11.3137	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$104$	−3.27620	−0.321258
$105$	0	0
$106$	8.00000	0.777029
$107$	−5.63325	−0.544587	−0.272293	−	0.962214i	$-0.587782\pi$
$107$	−5.63325	−0.544587	−0.272293	+	0.962214i	$0.587782\pi$
$108$	10.1234	0.974123
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	14.1421	1.34231
$112$	0	0
$113$	−0.683375	−0.0642865	−0.0321433	−	0.999483i	$-0.510233\pi$
$113$	−0.683375	−0.0642865	−0.0321433	+	0.999483i	$0.510233\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	−2.63325	−0.244491
$117$	20.6945	1.91321
$118$	−1.86199	−0.171410
$119$	0	0
$120$	0	0
$121$	17.2665	1.56968
$122$	−6.10463	−0.552687
$123$	7.94987	0.716816
$124$	−5.65685	−0.508001
$125$	0	0
$126$	0	0
$127$	18.6332	1.65343	0.826717	−	0.562618i	$-0.190206\pi$
$127$	18.6332	1.65343	0.826717	+	0.562618i	$0.190206\pi$
$128$	−1.00000	−0.0883883
$129$	−20.2468	−1.78263
$130$	0	0
$131$	−12.2801	−1.07292	−0.536461	−	0.843925i	$-0.680239\pi$
$131$	−12.2801	−1.07292	−0.536461	+	0.843925i	$0.680239\pi$
$132$	16.2280	1.41247
$133$	0	0
$134$	1.63325	0.141091
$135$	0	0
$136$	7.29496	0.625537
$137$	−13.9499	−1.19182	−0.595909	−	0.803052i	$-0.703208\pi$
$137$	−13.9499	−1.19182	−0.595909	+	0.803052i	$0.703208\pi$
$138$	−26.3514	−2.24318
$139$	−15.7093	−1.33245	−0.666225	−	0.745751i	$-0.732091\pi$
$139$	−15.7093	−1.33245	−0.666225	+	0.745751i	$0.732091\pi$
$140$	0	0
$141$	−18.6332	−1.56920
$142$	2.00000	0.167836
$143$	17.4183	1.45659
$144$	6.31662	0.526385
$145$	0	0
$146$	7.29496	0.603735
$147$	0	0
$148$	4.63325	0.380851
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−0.633250	−0.0515331	−0.0257666	−	0.999668i	$-0.508203\pi$
$151$	−0.633250	−0.0515331	−0.0257666	+	0.999668i	$0.508203\pi$
$152$	−1.63810	−0.132868
$153$	−46.0795	−3.72531
$154$	0	0
$155$	0	0
$156$	10.0000	0.800641
$157$	−9.38083	−0.748672	−0.374336	−	0.927293i	$-0.622129\pi$
$157$	−9.38083	−0.748672	−0.374336	+	0.927293i	$0.622129\pi$
$158$	−1.36675	−0.108733
$159$	−24.4185	−1.93651
$160$	0	0
$161$	0	0
$162$	−11.9499	−0.938871
$163$	2.68338	0.210178	0.105089	−	0.994463i	$-0.466487\pi$
$163$	2.68338	0.210178	0.105089	+	0.994463i	$0.466487\pi$
$164$	2.60454	0.203380
$165$	0	0
$166$	8.26139	0.641209
$167$	−21.5901	−1.67069	−0.835346	−	0.549725i	$-0.814732\pi$
$167$	−21.5901	−1.67069	−0.835346	+	0.549725i	$0.814732\pi$
$168$	0	0
$169$	−2.26650	−0.174346
$170$	0	0
$171$	10.3473	0.791276
$172$	−6.63325	−0.505781
$173$	−7.00018	−0.532214	−0.266107	−	0.963944i	$-0.585737\pi$
$173$	−7.00018	−0.532214	−0.266107	+	0.963944i	$0.585737\pi$
$174$	8.03751	0.609322
$175$	0	0
$176$	5.31662	0.400756
$177$	5.68338	0.427189
$178$	−4.91430	−0.368343
$179$	20.8997	1.56212	0.781060	−	0.624456i	$-0.214679\pi$
$179$	20.8997	1.56212	0.781060	+	0.624456i	$0.214679\pi$
$180$	0	0
$181$	9.82861	0.730555	0.365277	−	0.930899i	$-0.380974\pi$
$181$	9.82861	0.730555	0.365277	+	0.930899i	$0.380974\pi$
$182$	0	0
$183$	18.6332	1.37741
$184$	−8.63325	−0.636452
$185$	0	0
$186$	17.2665	1.26604
$187$	−38.7845	−2.83621
$188$	−6.10463	−0.445226
$189$	0	0
$190$	0	0
$191$	15.2665	1.10465	0.552323	−	0.833630i	$-0.313742\pi$
$191$	15.2665	1.10465	0.552323	+	0.833630i	$0.313742\pi$
$192$	3.05231	0.220282
$193$	3.31662	0.238736	0.119368	−	0.992850i	$-0.461913\pi$
$193$	3.31662	0.238736	0.119368	+	0.992850i	$0.461913\pi$
$194$	−4.24264	−0.304604
$195$	0	0
$196$	0	0
$197$	−6.63325	−0.472599	−0.236300	−	0.971680i	$-0.575935\pi$
$197$	−6.63325	−0.472599	−0.236300	+	0.971680i	$0.575935\pi$
$198$	−33.5831	−2.38665
$199$	−8.93306	−0.633248	−0.316624	−	0.948551i	$-0.602549\pi$
$199$	−8.93306	−0.633248	−0.316624	+	0.948551i	$0.602549\pi$
$200$	0	0
$201$	−4.98519	−0.351628
$202$	−15.0377	−1.05805
$203$	0	0
$204$	−22.2665	−1.55897
$205$	0	0
$206$	−11.3137	−0.788263
$207$	54.5330	3.79031
$208$	3.27620	0.227164
$209$	8.70917	0.602426
$210$	0	0
$211$	−1.63325	−0.112438	−0.0562188	−	0.998418i	$-0.517904\pi$
$211$	−1.63325	−0.112438	−0.0562188	+	0.998418i	$0.517904\pi$
$212$	−8.00000	−0.549442
$213$	−6.10463	−0.418282
$214$	5.63325	0.385081
$215$	0	0
$216$	−10.1234	−0.688809
$217$	0	0
$218$	−18.0000	−1.21911
$219$	−22.2665	−1.50463
$220$	0	0
$221$	−23.8997	−1.60767
$222$	−14.1421	−0.949158
$223$	3.27620	0.219391	0.109695	−	0.993965i	$-0.465012\pi$
$223$	3.27620	0.219391	0.109695	+	0.993965i	$0.465012\pi$
$224$	0	0
$225$	0	0
$226$	0.683375	0.0454574
$227$	3.79487	0.251874	0.125937	−	0.992038i	$-0.459806\pi$
$227$	3.79487	0.251874	0.125937	+	0.992038i	$0.459806\pi$
$228$	5.00000	0.331133
$229$	14.5899	0.964128	0.482064	−	0.876136i	$-0.339887\pi$
$229$	14.5899	0.964128	0.482064	+	0.876136i	$0.339887\pi$
$230$	0	0
$231$	0	0
$232$	2.63325	0.172881
$233$	−13.2665	−0.869117	−0.434559	−	0.900644i	$-0.643096\pi$
$233$	−13.2665	−0.869117	−0.434559	+	0.900644i	$0.643096\pi$
$234$	−20.6945	−1.35284
$235$	0	0
$236$	1.86199	0.121205
$237$	4.17175	0.270984
$238$	0	0
$239$	−16.6332	−1.07592	−0.537958	−	0.842972i	$-0.680804\pi$
$239$	−16.6332	−1.07592	−0.537958	+	0.842972i	$0.680804\pi$
$240$	0	0
$241$	−21.8140	−1.40516	−0.702581	−	0.711604i	$-0.747969\pi$
$241$	−21.8140	−1.40516	−0.702581	+	0.711604i	$0.747969\pi$
$242$	−17.2665	−1.10993
$243$	6.10463	0.391612
$244$	6.10463	0.390809
$245$	0	0
$246$	−7.94987	−0.506865
$247$	5.36675	0.341478
$248$	5.65685	0.359211
$249$	−25.2164	−1.59802
$250$	0	0
$251$	18.6087	1.17457	0.587284	−	0.809381i	$-0.300197\pi$
$251$	18.6087	1.17457	0.587284	+	0.809381i	$0.300197\pi$
$252$	0	0
$253$	45.8997	2.88569
$254$	−18.6332	−1.16915
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.41421	0.0882162	0.0441081	−	0.999027i	$-0.485955\pi$
$257$	1.41421	0.0882162	0.0441081	+	0.999027i	$0.485955\pi$
$258$	20.2468	1.26051
$259$	0	0
$260$	0	0
$261$	−16.6332	−1.02957
$262$	12.2801	0.758670
$263$	−15.8997	−0.980421	−0.490210	−	0.871604i	$-0.663080\pi$
$263$	−15.8997	−0.980421	−0.490210	+	0.871604i	$0.663080\pi$
$264$	−16.2280	−0.998765
$265$	0	0
$266$	0	0
$267$	15.0000	0.917985
$268$	−1.63325	−0.0997666
$269$	−8.03751	−0.490055	−0.245028	−	0.969516i	$-0.578797\pi$
$269$	−8.03751	−0.490055	−0.245028	+	0.969516i	$0.578797\pi$
$270$	0	0
$271$	−1.34333	−0.0816012	−0.0408006	−	0.999167i	$-0.512991\pi$
$271$	−1.34333	−0.0816012	−0.0408006	+	0.999167i	$0.512991\pi$
$272$	−7.29496	−0.442322
$273$	0	0
$274$	13.9499	0.842743
$275$	0	0
$276$	26.3514	1.58617
$277$	6.63325	0.398553	0.199277	−	0.979943i	$-0.436141\pi$
$277$	6.63325	0.398553	0.199277	+	0.979943i	$0.436141\pi$
$278$	15.7093	0.942184
$279$	−35.7322	−2.13923
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	18.6332	1.10959
$283$	−25.2320	−1.49988	−0.749942	−	0.661504i	$-0.769919\pi$
$283$	−25.2320	−1.49988	−0.749942	+	0.661504i	$0.769919\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−17.4183	−1.02997
$287$	0	0
$288$	−6.31662	−0.372211
$289$	36.2164	2.13037
$290$	0	0
$291$	12.9499	0.759135
$292$	−7.29496	−0.426905
$293$	−5.20908	−0.304318	−0.152159	−	0.988356i	$-0.548623\pi$
$293$	−5.20908	−0.304318	−0.152159	+	0.988356i	$0.548623\pi$
$294$	0	0
$295$	0	0
$296$	−4.63325	−0.269302
$297$	53.8222	3.12308
$298$	0	0
$299$	28.2843	1.63572
$300$	0	0
$301$	0	0
$302$	0.633250	0.0364394
$303$	45.8997	2.63687
$304$	1.63810	0.0939515
$305$	0	0
$306$	46.0795	2.63419
$307$	15.3325	0.875070	0.437535	−	0.899201i	$-0.355852\pi$
$307$	15.3325	0.875070	0.437535	+	0.899201i	$0.355852\pi$
$308$	0	0
$309$	34.5330	1.96451
$310$	0	0
$311$	−27.6947	−1.57042	−0.785212	−	0.619227i	$-0.787446\pi$
$311$	−27.6947	−1.57042	−0.785212	+	0.619227i	$0.787446\pi$
$312$	−10.0000	−0.566139
$313$	−11.6906	−0.660792	−0.330396	−	0.943842i	$-0.607182\pi$
$313$	−11.6906	−0.660792	−0.330396	+	0.943842i	$0.607182\pi$
$314$	9.38083	0.529391
$315$	0	0
$316$	1.36675	0.0768857
$317$	33.8997	1.90400	0.952000	−	0.306099i	$-0.0990238\pi$
$317$	33.8997	1.90400	0.952000	+	0.306099i	$0.0990238\pi$
$318$	24.4185	1.36932
$319$	−14.0000	−0.783850
$320$	0	0
$321$	−17.1945	−0.959701
$322$	0	0
$323$	−11.9499	−0.664909
$324$	11.9499	0.663882
$325$	0	0
$326$	−2.68338	−0.148618
$327$	54.9417	3.03828
$328$	−2.60454	−0.143812
$329$	0	0
$330$	0	0
$331$	22.6834	1.24679	0.623396	−	0.781907i	$-0.285753\pi$
$331$	22.6834	1.24679	0.623396	+	0.781907i	$0.285753\pi$
$332$	−8.26139	−0.453403
$333$	29.2665	1.60379
$334$	21.5901	1.18136
$335$	0	0
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	2.26650	0.123281
$339$	−2.08588	−0.113289
$340$	0	0
$341$	−30.0754	−1.62867
$342$	−10.3473	−0.559516
$343$	0	0
$344$	6.63325	0.357641
$345$	0	0
$346$	7.00018	0.376332
$347$	2.68338	0.144051	0.0720256	−	0.997403i	$-0.477054\pi$
$347$	2.68338	0.144051	0.0720256	+	0.997403i	$0.477054\pi$
$348$	−8.03751	−0.430856
$349$	19.6572	1.05223	0.526113	−	0.850415i	$-0.323649\pi$
$349$	19.6572	1.05223	0.526113	+	0.850415i	$0.323649\pi$
$350$	0	0
$351$	33.1662	1.77028
$352$	−5.31662	−0.283377
$353$	9.00394	0.479232	0.239616	−	0.970868i	$-0.422979\pi$
$353$	9.00394	0.479232	0.239616	+	0.970868i	$0.422979\pi$
$354$	−5.68338	−0.302068
$355$	0	0
$356$	4.91430	0.260458
$357$	0	0
$358$	−20.8997	−1.10459
$359$	−31.2665	−1.65018	−0.825091	−	0.564999i	$-0.808876\pi$
$359$	−31.2665	−1.65018	−0.825091	+	0.564999i	$0.808876\pi$
$360$	0	0
$361$	−16.3166	−0.858770
$362$	−9.82861	−0.516580
$363$	52.7028	2.76618
$364$	0	0
$365$	0	0
$366$	−18.6332	−0.973976
$367$	−27.2469	−1.42228	−0.711139	−	0.703051i	$-0.751821\pi$
$367$	−27.2469	−1.42228	−0.711139	+	0.703051i	$0.751821\pi$
$368$	8.63325	0.450039
$369$	16.4519	0.856452
$370$	0	0
$371$	0	0
$372$	−17.2665	−0.895226
$373$	33.8997	1.75526	0.877631	−	0.479336i	$-0.159123\pi$
$373$	33.8997	1.75526	0.877631	+	0.479336i	$0.159123\pi$
$374$	38.7845	2.00550
$375$	0	0
$376$	6.10463	0.314822
$377$	−8.62706	−0.444316
$378$	0	0
$379$	−7.94987	−0.408358	−0.204179	−	0.978934i	$-0.565452\pi$
$379$	−7.94987	−0.408358	−0.204179	+	0.978934i	$0.565452\pi$
$380$	0	0
$381$	56.8745	2.91377
$382$	−15.2665	−0.781102
$383$	18.3139	0.935796	0.467898	−	0.883782i	$-0.345011\pi$
$383$	18.3139	0.935796	0.467898	+	0.883782i	$0.345011\pi$
$384$	−3.05231	−0.155763
$385$	0	0
$386$	−3.31662	−0.168812
$387$	−41.8997	−2.12988
$388$	4.24264	0.215387
$389$	−28.6332	−1.45176	−0.725882	−	0.687820i	$-0.758568\pi$
$389$	−28.6332	−1.45176	−0.725882	+	0.687820i	$0.758568\pi$
$390$	0	0
$391$	−62.9792	−3.18499
$392$	0	0
$393$	−37.4829	−1.89076
$394$	6.63325	0.334178
$395$	0	0
$396$	33.5831	1.68762
$397$	−11.7615	−0.590292	−0.295146	−	0.955452i	$-0.595368\pi$
$397$	−11.7615	−0.590292	−0.295146	+	0.955452i	$0.595368\pi$
$398$	8.93306	0.447774
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	4.98519	0.248639
$403$	−18.5330	−0.923194
$404$	15.0377	0.748153
$405$	0	0
$406$	0	0
$407$	24.6332	1.22102
$408$	22.2665	1.10236
$409$	38.7845	1.91777	0.958886	−	0.283791i	$-0.0915923\pi$
$409$	38.7845	1.91777	0.958886	+	0.283791i	$0.0915923\pi$
$410$	0	0
$411$	−42.5794	−2.10029
$412$	11.3137	0.557386
$413$	0	0
$414$	−54.5330	−2.68015
$415$	0	0
$416$	−3.27620	−0.160629
$417$	−47.9499	−2.34812
$418$	−8.70917	−0.425979
$419$	−19.1273	−0.934431	−0.467216	−	0.884143i	$-0.654743\pi$
$419$	−19.1273	−0.934431	−0.467216	+	0.884143i	$0.654743\pi$
$420$	0	0
$421$	38.5330	1.87798	0.938992	−	0.343940i	$-0.111762\pi$
$421$	38.5330	1.87798	0.938992	+	0.343940i	$0.111762\pi$
$422$	1.63325	0.0795053
$423$	−38.5607	−1.87488
$424$	8.00000	0.388514
$425$	0	0
$426$	6.10463	0.295770
$427$	0	0
$428$	−5.63325	−0.272293
$429$	53.1662	2.56689
$430$	0	0
$431$	7.36675	0.354844	0.177422	−	0.984135i	$-0.443224\pi$
$431$	7.36675	0.354844	0.177422	+	0.984135i	$0.443224\pi$
$432$	10.1234	0.487061
$433$	−3.50009	−0.168204	−0.0841018	−	0.996457i	$-0.526802\pi$
$433$	−3.50009	−0.168204	−0.0841018	+	0.996457i	$0.526802\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	14.1421	0.676510
$438$	22.2665	1.06393
$439$	−9.38083	−0.447723	−0.223861	−	0.974621i	$-0.571866\pi$
$439$	−9.38083	−0.447723	−0.223861	+	0.974621i	$0.571866\pi$
$440$	0	0
$441$	0	0
$442$	23.8997	1.13680
$443$	12.8997	0.612886	0.306443	−	0.951889i	$-0.400861\pi$
$443$	12.8997	0.612886	0.306443	+	0.951889i	$0.400861\pi$
$444$	14.1421	0.671156
$445$	0	0
$446$	−3.27620	−0.155133
$447$	0	0
$448$	0	0
$449$	−34.2665	−1.61714	−0.808568	−	0.588403i	$-0.799757\pi$
$449$	−34.2665	−1.61714	−0.808568	+	0.588403i	$0.799757\pi$
$450$	0	0
$451$	13.8474	0.652047
$452$	−0.683375	−0.0321433
$453$	−1.93288	−0.0908145
$454$	−3.79487	−0.178102
$455$	0	0
$456$	−5.00000	−0.234146
$457$	−16.6834	−0.780415	−0.390208	−	0.920727i	$-0.627597\pi$
$457$	−16.6834	−0.780415	−0.390208	+	0.920727i	$0.627597\pi$
$458$	−14.5899	−0.681742
$459$	−73.8496	−3.44701
$460$	0	0
$461$	−6.10463	−0.284321	−0.142160	−	0.989844i	$-0.545405\pi$
$461$	−6.10463	−0.284321	−0.142160	+	0.989844i	$0.545405\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−2.63325	−0.122246
$465$	0	0
$466$	13.2665	0.614559
$467$	0.0708883	0.00328032	0.00164016	−	0.999999i	$-0.499478\pi$
$467$	0.0708883	0.00328032	0.00164016	+	0.999999i	$0.499478\pi$
$468$	20.6945	0.956605
$469$	0	0
$470$	0	0
$471$	−28.6332	−1.31935
$472$	−1.86199	−0.0857050
$473$	−35.2665	−1.62156
$474$	−4.17175	−0.191615
$475$	0	0
$476$	0	0
$477$	−50.5330	−2.31375
$478$	16.6332	0.760787
$479$	−5.06730	−0.231531	−0.115765	−	0.993277i	$-0.536932\pi$
$479$	−5.06730	−0.231531	−0.115765	+	0.993277i	$0.536932\pi$
$480$	0	0
$481$	15.1795	0.692124
$482$	21.8140	0.993599
$483$	0	0
$484$	17.2665	0.784841
$485$	0	0
$486$	−6.10463	−0.276912
$487$	−31.2665	−1.41682	−0.708410	−	0.705801i	$-0.750587\pi$
$487$	−31.2665	−1.41682	−0.708410	+	0.705801i	$0.750587\pi$
$488$	−6.10463	−0.276344
$489$	8.19051	0.370387
$490$	0	0
$491$	−21.2665	−0.959744	−0.479872	−	0.877339i	$-0.659317\pi$
$491$	−21.2665	−0.959744	−0.479872	+	0.877339i	$0.659317\pi$
$492$	7.94987	0.358408
$493$	19.2094	0.865150
$494$	−5.36675	−0.241462
$495$	0	0
$496$	−5.65685	−0.254000
$497$	0	0
$498$	25.2164	1.12997
$499$	−13.2665	−0.593890	−0.296945	−	0.954895i	$-0.595968\pi$
$499$	−13.2665	−0.593890	−0.296945	+	0.954895i	$0.595968\pi$
$500$	0	0
$501$	−65.8997	−2.94418
$502$	−18.6087	−0.830545
$503$	12.6570	0.564349	0.282175	−	0.959363i	$-0.408944\pi$
$503$	12.6570	0.564349	0.282175	+	0.959363i	$0.408944\pi$
$504$	0	0
$505$	0	0
$506$	−45.8997	−2.04049
$507$	−6.91807	−0.307242
$508$	18.6332	0.826717
$509$	−2.38065	−0.105521	−0.0527603	−	0.998607i	$-0.516802\pi$
$509$	−2.38065	−0.105521	−0.0527603	+	0.998607i	$0.516802\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	16.5831	0.732163
$514$	−1.41421	−0.0623783
$515$	0	0
$516$	−20.2468	−0.891314
$517$	−32.4560	−1.42741
$518$	0	0
$519$	−21.3668	−0.937896
$520$	0	0
$521$	−40.1279	−1.75803	−0.879017	−	0.476791i	$-0.841800\pi$
$521$	−40.1279	−1.75803	−0.879017	+	0.476791i	$0.841800\pi$
$522$	16.6332	0.728018
$523$	34.0941	1.49083	0.745416	−	0.666600i	$-0.232251\pi$
$523$	34.0941	1.49083	0.745416	+	0.666600i	$0.232251\pi$
$524$	−12.2801	−0.536461
$525$	0	0
$526$	15.8997	0.693262
$527$	41.2665	1.79760
$528$	16.2280	0.706234
$529$	51.5330	2.24057
$530$	0	0
$531$	11.7615	0.510405
$532$	0	0
$533$	8.53300	0.369605
$534$	−15.0000	−0.649113
$535$	0	0
$536$	1.63325	0.0705456
$537$	63.7926	2.75285
$538$	8.03751	0.346522
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	1.34333	0.0577008
$543$	30.0000	1.28742
$544$	7.29496	0.312769
$545$	0	0
$546$	0	0
$547$	−35.9499	−1.53711	−0.768553	−	0.639786i	$-0.779023\pi$
$547$	−35.9499	−1.53711	−0.768553	+	0.639786i	$0.779023\pi$
$548$	−13.9499	−0.595909
$549$	38.5607	1.64573
$550$	0	0
$551$	−4.31353	−0.183763
$552$	−26.3514	−1.12159
$553$	0	0
$554$	−6.63325	−0.281820
$555$	0	0
$556$	−15.7093	−0.666225
$557$	−10.7335	−0.454793	−0.227397	−	0.973802i	$-0.573021\pi$
$557$	−10.7335	−0.454793	−0.227397	+	0.973802i	$0.573021\pi$
$558$	35.7322	1.51267
$559$	−21.7319	−0.919160
$560$	0	0
$561$	−118.383	−4.99812
$562$	4.00000	0.168730
$563$	−9.45172	−0.398342	−0.199171	−	0.979965i	$-0.563825\pi$
$563$	−9.45172	−0.398342	−0.199171	+	0.979965i	$0.563825\pi$
$564$	−18.6332	−0.784601
$565$	0	0
$566$	25.2320	1.06058
$567$	0	0
$568$	2.00000	0.0839181
$569$	4.26650	0.178861	0.0894305	−	0.995993i	$-0.471495\pi$
$569$	4.26650	0.178861	0.0894305	+	0.995993i	$0.471495\pi$
$570$	0	0
$571$	2.63325	0.110198	0.0550990	−	0.998481i	$-0.482453\pi$
$571$	2.63325	0.110198	0.0550990	+	0.998481i	$0.482453\pi$
$572$	17.4183	0.728297
$573$	46.5982	1.94667
$574$	0	0
$575$	0	0
$576$	6.31662	0.263193
$577$	36.4748	1.51846	0.759232	−	0.650820i	$-0.225575\pi$
$577$	36.4748	1.51846	0.759232	+	0.650820i	$0.225575\pi$
$578$	−36.2164	−1.50640
$579$	10.1234	0.420713
$580$	0	0
$581$	0	0
$582$	−12.9499	−0.536790
$583$	−42.5330	−1.76154
$584$	7.29496	0.301867
$585$	0	0
$586$	5.20908	0.215185
$587$	−17.1236	−0.706765	−0.353383	−	0.935479i	$-0.614969\pi$
$587$	−17.1236	−0.706765	−0.353383	+	0.935479i	$0.614969\pi$
$588$	0	0
$589$	−9.26650	−0.381819
$590$	0	0
$591$	−20.2468	−0.832841
$592$	4.63325	0.190425
$593$	−29.9224	−1.22876	−0.614382	−	0.789009i	$-0.710595\pi$
$593$	−29.9224	−1.22876	−0.614382	+	0.789009i	$0.710595\pi$
$594$	−53.8222	−2.20835
$595$	0	0
$596$	0	0
$597$	−27.2665	−1.11594
$598$	−28.2843	−1.15663
$599$	−6.73350	−0.275123	−0.137562	−	0.990493i	$-0.543927\pi$
$599$	−6.73350	−0.275123	−0.137562	+	0.990493i	$0.543927\pi$
$600$	0	0
$601$	−1.26121	−0.0514460	−0.0257230	−	0.999669i	$-0.508189\pi$
$601$	−1.26121	−0.0514460	−0.0257230	+	0.999669i	$0.508189\pi$
$602$	0	0
$603$	−10.3166	−0.420125
$604$	−0.633250	−0.0257666
$605$	0	0
$606$	−45.8997	−1.86455
$607$	−11.3137	−0.459209	−0.229605	−	0.973284i	$-0.573743\pi$
$607$	−11.3137	−0.459209	−0.229605	+	0.973284i	$0.573743\pi$
$608$	−1.63810	−0.0664338
$609$	0	0
$610$	0	0
$611$	−20.0000	−0.809113
$612$	−46.0795	−1.86265
$613$	5.36675	0.216761	0.108381	−	0.994109i	$-0.465434\pi$
$613$	5.36675	0.216761	0.108381	+	0.994109i	$0.465434\pi$
$614$	−15.3325	−0.618768
$615$	0	0
$616$	0	0
$617$	15.2665	0.614606	0.307303	−	0.951612i	$-0.400573\pi$
$617$	15.2665	0.614606	0.307303	+	0.951612i	$0.400573\pi$
$618$	−34.5330	−1.38912
$619$	−26.2805	−1.05630	−0.528151	−	0.849150i	$-0.677115\pi$
$619$	−26.2805	−1.05630	−0.528151	+	0.849150i	$0.677115\pi$
$620$	0	0
$621$	87.3977	3.50715
$622$	27.6947	1.11046
$623$	0	0
$624$	10.0000	0.400320
$625$	0	0
$626$	11.6906	0.467250
$627$	26.5831	1.06163
$628$	−9.38083	−0.374336
$629$	−33.7993	−1.34767
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	−1.36675	−0.0543664
$633$	−4.98519	−0.198144
$634$	−33.8997	−1.34633
$635$	0	0
$636$	−24.4185	−0.968257
$637$	0	0
$638$	14.0000	0.554265
$639$	−12.6332	−0.499764
$640$	0	0
$641$	24.5330	0.968995	0.484498	−	0.874793i	$-0.339002\pi$
$641$	24.5330	0.968995	0.484498	+	0.874793i	$0.339002\pi$
$642$	17.1945	0.678611
$643$	29.2507	1.15354	0.576768	−	0.816908i	$-0.304314\pi$
$643$	29.2507	1.15354	0.576768	+	0.816908i	$0.304314\pi$
$644$	0	0
$645$	0	0
$646$	11.9499	0.470162
$647$	−21.5901	−0.848794	−0.424397	−	0.905476i	$-0.639514\pi$
$647$	−21.5901	−0.848794	−0.424397	+	0.905476i	$0.639514\pi$
$648$	−11.9499	−0.469435
$649$	9.89949	0.388589
$650$	0	0
$651$	0	0
$652$	2.68338	0.105089
$653$	−35.1662	−1.37616	−0.688081	−	0.725634i	$-0.741547\pi$
$653$	−35.1662	−1.37616	−0.688081	+	0.725634i	$0.741547\pi$
$654$	−54.9417	−2.14839
$655$	0	0
$656$	2.60454	0.101690
$657$	−46.0795	−1.79773
$658$	0	0
$659$	−13.4169	−0.522647	−0.261324	−	0.965251i	$-0.584159\pi$
$659$	−13.4169	−0.522647	−0.261324	+	0.965251i	$0.584159\pi$
$660$	0	0
$661$	−23.0752	−0.897521	−0.448760	−	0.893652i	$-0.648134\pi$
$661$	−23.0752	−0.897521	−0.448760	+	0.893652i	$0.648134\pi$
$662$	−22.6834	−0.881614
$663$	−72.9496	−2.83313
$664$	8.26139	0.320604
$665$	0	0
$666$	−29.2665	−1.13405
$667$	−22.7335	−0.880245
$668$	−21.5901	−0.835346
$669$	10.0000	0.386622
$670$	0	0
$671$	32.4560	1.25295
$672$	0	0
$673$	12.0000	0.462566	0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000	0.462566	0.231283	+	0.972887i	$0.425708\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	−2.26650	−0.0871730
$677$	−10.2764	−0.394953	−0.197477	−	0.980308i	$-0.563275\pi$
$677$	−10.2764	−0.394953	−0.197477	+	0.980308i	$0.563275\pi$
$678$	2.08588	0.0801076
$679$	0	0
$680$	0	0
$681$	11.5831	0.443866
$682$	30.0754	1.15165
$683$	16.1662	0.618584	0.309292	−	0.950967i	$-0.399908\pi$
$683$	16.1662	0.618584	0.309292	+	0.950967i	$0.399908\pi$
$684$	10.3473	0.395638
$685$	0	0
$686$	0	0
$687$	44.5330	1.69904
$688$	−6.63325	−0.252890
$689$	−26.2096	−0.998507
$690$	0	0
$691$	38.8554	1.47813	0.739065	−	0.673634i	$-0.235268\pi$
$691$	38.8554	1.47813	0.739065	+	0.673634i	$0.235268\pi$
$692$	−7.00018	−0.266107
$693$	0	0
$694$	−2.68338	−0.101860
$695$	0	0
$696$	8.03751	0.304661
$697$	−19.0000	−0.719676
$698$	−19.6572	−0.744036
$699$	−40.4935	−1.53161
$700$	0	0
$701$	27.7995	1.04997	0.524986	−	0.851111i	$-0.324070\pi$
$701$	27.7995	1.04997	0.524986	+	0.851111i	$0.324070\pi$
$702$	−33.1662	−1.25178
$703$	7.58973	0.286252
$704$	5.31662	0.200378
$705$	0	0
$706$	−9.00394	−0.338868
$707$	0	0
$708$	5.68338	0.213594
$709$	−18.5330	−0.696021	−0.348011	−	0.937491i	$-0.613143\pi$
$709$	−18.5330	−0.696021	−0.348011	+	0.937491i	$0.613143\pi$
$710$	0	0
$711$	8.63325	0.323772
$712$	−4.91430	−0.184171
$713$	−48.8370	−1.82896
$714$	0	0
$715$	0	0
$716$	20.8997	0.781060
$717$	−50.7699	−1.89604
$718$	31.2665	1.16686
$719$	−17.8661	−0.666294	−0.333147	−	0.942875i	$-0.608110\pi$
$719$	−17.8661	−0.666294	−0.333147	+	0.942875i	$0.608110\pi$
$720$	0	0
$721$	0	0
$722$	16.3166	0.607242
$723$	−66.5831	−2.47625
$724$	9.82861	0.365277
$725$	0	0
$726$	−52.7028	−1.95598
$727$	43.7697	1.62333	0.811665	−	0.584124i	$-0.198562\pi$
$727$	43.7697	1.62333	0.811665	+	0.584124i	$0.198562\pi$
$728$	0	0
$729$	−17.2164	−0.637643
$730$	0	0
$731$	48.3893	1.78974
$732$	18.6332	0.688705
$733$	7.58973	0.280333	0.140167	−	0.990128i	$-0.455236\pi$
$733$	7.58973	0.280333	0.140167	+	0.990128i	$0.455236\pi$
$734$	27.2469	1.00570
$735$	0	0
$736$	−8.63325	−0.318226
$737$	−8.68338	−0.319856
$738$	−16.4519	−0.605603
$739$	33.1662	1.22004	0.610020	−	0.792386i	$-0.291161\pi$
$739$	33.1662	1.22004	0.610020	+	0.792386i	$0.291161\pi$
$740$	0	0
$741$	16.3810	0.601771
$742$	0	0
$743$	8.63325	0.316723	0.158362	−	0.987381i	$-0.449379\pi$
$743$	8.63325	0.316723	0.158362	+	0.987381i	$0.449379\pi$
$744$	17.2665	0.633021
$745$	0	0
$746$	−33.8997	−1.24116
$747$	−52.1841	−1.90932
$748$	−38.7845	−1.41810
$749$	0	0
$750$	0	0
$751$	1.36675	0.0498734	0.0249367	−	0.999689i	$-0.492062\pi$
$751$	1.36675	0.0498734	0.0249367	+	0.999689i	$0.492062\pi$
$752$	−6.10463	−0.222613
$753$	56.7995	2.06989
$754$	8.62706	0.314179
$755$	0	0
$756$	0	0
$757$	−26.6332	−0.968002	−0.484001	−	0.875067i	$-0.660817\pi$
$757$	−26.6332	−0.968002	−0.484001	+	0.875067i	$0.660817\pi$
$758$	7.94987	0.288752
$759$	140.100	5.08533
$760$	0	0
$761$	−32.6799	−1.18465	−0.592323	−	0.805701i	$-0.701789\pi$
$761$	−32.6799	−1.18465	−0.592323	+	0.805701i	$0.701789\pi$
$762$	−56.8745	−2.06035
$763$	0	0
$764$	15.2665	0.552323
$765$	0	0
$766$	−18.3139	−0.661708
$767$	6.10025	0.220267
$768$	3.05231	0.110141
$769$	−36.7808	−1.32635	−0.663174	−	0.748465i	$-0.730791\pi$
$769$	−36.7808	−1.32635	−0.663174	+	0.748465i	$0.730791\pi$
$770$	0	0
$771$	4.31662	0.155459
$772$	3.31662	0.119368
$773$	35.7322	1.28520	0.642599	−	0.766202i	$-0.277856\pi$
$773$	35.7322	1.28520	0.642599	+	0.766202i	$0.277856\pi$
$774$	41.8997	1.50606
$775$	0	0
$776$	−4.24264	−0.152302
$777$	0	0
$778$	28.6332	1.02655
$779$	4.26650	0.152863
$780$	0	0
$781$	−10.6332	−0.380488
$782$	62.9792	2.25213
$783$	−26.6574	−0.952657
$784$	0	0
$785$	0	0
$786$	37.4829	1.33697
$787$	−37.7360	−1.34514	−0.672571	−	0.740032i	$-0.734810\pi$
$787$	−37.7360	−1.34514	−0.672571	+	0.740032i	$0.734810\pi$
$788$	−6.63325	−0.236300
$789$	−48.5310	−1.72775
$790$	0	0
$791$	0	0
$792$	−33.5831	−1.19332
$793$	20.0000	0.710221
$794$	11.7615	0.417399
$795$	0	0
$796$	−8.93306	−0.316624
$797$	−31.5605	−1.11793	−0.558965	−	0.829191i	$-0.688801\pi$
$797$	−31.5605	−1.11793	−0.558965	+	0.829191i	$0.688801\pi$
$798$	0	0
$799$	44.5330	1.57546
$800$	0	0
$801$	31.0418	1.09681
$802$	3.00000	0.105934
$803$	−38.7845	−1.36868
$804$	−4.98519	−0.175814
$805$	0	0
$806$	18.5330	0.652797
$807$	−24.5330	−0.863602
$808$	−15.0377	−0.529024
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	1.86199	0.0653833	0.0326916	−	0.999465i	$-0.489592\pi$
$811$	1.86199	0.0653833	0.0326916	+	0.999465i	$0.489592\pi$
$812$	0	0
$813$	−4.10025	−0.143802
$814$	−24.6332	−0.863395
$815$	0	0
$816$	−22.2665	−0.779483
$817$	−10.8659	−0.380151
$818$	−38.7845	−1.35607
$819$	0	0
$820$	0	0
$821$	22.7335	0.793405	0.396702	−	0.917947i	$-0.370155\pi$
$821$	22.7335	0.793405	0.396702	+	0.917947i	$0.370155\pi$
$822$	42.5794	1.48513
$823$	48.5330	1.69175	0.845877	−	0.533378i	$-0.179078\pi$
$823$	48.5330	1.69175	0.845877	+	0.533378i	$0.179078\pi$
$824$	−11.3137	−0.394132
$825$	0	0
$826$	0	0
$827$	−7.63325	−0.265434	−0.132717	−	0.991154i	$-0.542370\pi$
$827$	−7.63325	−0.265434	−0.132717	+	0.991154i	$0.542370\pi$
$828$	54.5330	1.89515
$829$	−44.2175	−1.53574	−0.767869	−	0.640607i	$-0.778683\pi$
$829$	−44.2175	−1.53574	−0.767869	+	0.640607i	$0.778683\pi$
$830$	0	0
$831$	20.2468	0.702352
$832$	3.27620	0.113582
$833$	0	0
$834$	47.9499	1.66037
$835$	0	0
$836$	8.70917	0.301213
$837$	−57.2665	−1.97942
$838$	19.1273	0.660743
$839$	−11.7615	−0.406052	−0.203026	−	0.979173i	$-0.565078\pi$
$839$	−11.7615	−0.406052	−0.203026	+	0.979173i	$0.565078\pi$
$840$	0	0
$841$	−22.0660	−0.760896
$842$	−38.5330	−1.32793
$843$	−12.2093	−0.420509
$844$	−1.63325	−0.0562188
$845$	0	0
$846$	38.5607	1.32574
$847$	0	0
$848$	−8.00000	−0.274721
$849$	−77.0159	−2.64318
$850$	0	0
$851$	40.0000	1.37118
$852$	−6.10463	−0.209141
$853$	−45.7026	−1.56483	−0.782414	−	0.622759i	$-0.786012\pi$
$853$	−45.7026	−1.56483	−0.782414	+	0.622759i	$0.786012\pi$
$854$	0	0
$855$	0	0
$856$	5.63325	0.192541
$857$	−4.84341	−0.165448	−0.0827240	−	0.996573i	$-0.526362\pi$
$857$	−4.84341	−0.165448	−0.0827240	+	0.996573i	$0.526362\pi$
$858$	−53.1662	−1.81507
$859$	27.0231	0.922015	0.461007	−	0.887396i	$-0.347488\pi$
$859$	27.0231	0.922015	0.461007	+	0.887396i	$0.347488\pi$
$860$	0	0
$861$	0	0
$862$	−7.36675	−0.250913
$863$	10.1003	0.343817	0.171908	−	0.985113i	$-0.445007\pi$
$863$	10.1003	0.343817	0.171908	+	0.985113i	$0.445007\pi$
$864$	−10.1234	−0.344404
$865$	0	0
$866$	3.50009	0.118938
$867$	110.544	3.75426
$868$	0	0
$869$	7.26650	0.246499
$870$	0	0
$871$	−5.35086	−0.181307
$872$	−18.0000	−0.609557
$873$	26.7992	0.907014
$874$	−14.1421	−0.478365
$875$	0	0
$876$	−22.2665	−0.752315
$877$	5.26650	0.177837	0.0889185	−	0.996039i	$-0.471659\pi$
$877$	5.26650	0.177837	0.0889185	+	0.996039i	$0.471659\pi$
$878$	9.38083	0.316588
$879$	−15.8997	−0.536285
$880$	0	0
$881$	−53.0797	−1.78830	−0.894150	−	0.447768i	$-0.852219\pi$
$881$	−53.0797	−1.78830	−0.894150	+	0.447768i	$0.852219\pi$
$882$	0	0
$883$	15.6332	0.526101	0.263050	−	0.964782i	$-0.415271\pi$
$883$	15.6332	0.526101	0.263050	+	0.964782i	$0.415271\pi$
$884$	−23.8997	−0.803836
$885$	0	0
$886$	−12.8997	−0.433376
$887$	−29.6276	−0.994797	−0.497399	−	0.867522i	$-0.665711\pi$
$887$	−29.6276	−0.994797	−0.497399	+	0.867522i	$0.665711\pi$
$888$	−14.1421	−0.474579
$889$	0	0
$890$	0	0
$891$	63.5330	2.12844
$892$	3.27620	0.109695
$893$	−10.0000	−0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	86.3325	2.88256
$898$	34.2665	1.14349
$899$	14.8959	0.496806
$900$	0	0
$901$	58.3596	1.94424
$902$	−13.8474	−0.461067
$903$	0	0
$904$	0.683375	0.0227287
$905$	0	0
$906$	1.93288	0.0642155
$907$	−5.26650	−0.174871	−0.0874356	−	0.996170i	$-0.527867\pi$
$907$	−5.26650	−0.174871	−0.0874356	+	0.996170i	$0.527867\pi$
$908$	3.79487	0.125937
$909$	94.9874	3.15053
$910$	0	0
$911$	−43.1662	−1.43016	−0.715081	−	0.699042i	$-0.753610\pi$
$911$	−43.1662	−1.43016	−0.715081	+	0.699042i	$0.753610\pi$
$912$	5.00000	0.165567
$913$	−43.9227	−1.45363
$914$	16.6834	0.551837
$915$	0	0
$916$	14.5899	0.482064
$917$	0	0
$918$	73.8496	2.43740
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	0	0
$921$	46.7995	1.54210
$922$	6.10463	0.201045
$923$	−6.55240	−0.215675
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	71.4645	2.34720
$928$	2.63325	0.0864407
$929$	−50.2512	−1.64869	−0.824345	−	0.566088i	$-0.808456\pi$
$929$	−50.2512	−1.64869	−0.824345	+	0.566088i	$0.808456\pi$
$930$	0	0
$931$	0	0
$932$	−13.2665	−0.434559
$933$	−84.5330	−2.76749
$934$	−0.0708883	−0.00231954
$935$	0	0
$936$	−20.6945	−0.676422
$937$	26.5753	0.868177	0.434088	−	0.900870i	$-0.357071\pi$
$937$	26.5753	0.868177	0.434088	+	0.900870i	$0.357071\pi$
$938$	0	0
$939$	−35.6834	−1.16448
$940$	0	0
$941$	52.7028	1.71806	0.859031	−	0.511924i	$-0.171067\pi$
$941$	52.7028	1.71806	0.859031	+	0.511924i	$0.171067\pi$
$942$	28.6332	0.932922
$943$	22.4856	0.732233
$944$	1.86199	0.0606026
$945$	0	0
$946$	35.2665	1.14661
$947$	33.3668	1.08427	0.542137	−	0.840290i	$-0.317615\pi$
$947$	33.3668	1.08427	0.542137	+	0.840290i	$0.317615\pi$
$948$	4.17175	0.135492
$949$	−23.8997	−0.775819
$950$	0	0
$951$	103.473	3.35533
$952$	0	0
$953$	−58.2665	−1.88744	−0.943719	−	0.330750i	$-0.892698\pi$
$953$	−58.2665	−1.88744	−0.943719	+	0.330750i	$0.892698\pi$
$954$	50.5330	1.63607
$955$	0	0
$956$	−16.6332	−0.537958
$957$	−42.7324	−1.38134
$958$	5.06730	0.163717
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−15.1795	−0.489406
$963$	−35.5831	−1.14665
$964$	−21.8140	−0.702581
$965$	0	0
$966$	0	0
$967$	−31.8997	−1.02583	−0.512913	−	0.858440i	$-0.671434\pi$
$967$	−31.8997	−1.02583	−0.512913	+	0.858440i	$0.671434\pi$
$968$	−17.2665	−0.554966
$969$	−36.4748	−1.17174
$970$	0	0
$971$	−15.3325	−0.492042	−0.246021	−	0.969264i	$-0.579123\pi$
$971$	−15.3325	−0.492042	−0.246021	+	0.969264i	$0.579123\pi$
$972$	6.10463	0.195806
$973$	0	0
$974$	31.2665	1.00184
$975$	0	0
$976$	6.10463	0.195404
$977$	20.6834	0.661720	0.330860	−	0.943680i	$-0.392661\pi$
$977$	20.6834	0.661720	0.330860	+	0.943680i	$0.392661\pi$
$978$	−8.19051	−0.261903
$979$	26.1275	0.835039
$980$	0	0
$981$	113.699	3.63014
$982$	21.2665	0.678641
$983$	49.8744	1.59075	0.795373	−	0.606121i	$-0.207275\pi$
$983$	49.8744	1.59075	0.795373	+	0.606121i	$0.207275\pi$
$984$	−7.94987	−0.253433
$985$	0	0
$986$	−19.2094	−0.611753
$987$	0	0
$988$	5.36675	0.170739
$989$	−57.2665	−1.82097
$990$	0	0
$991$	22.5330	0.715784	0.357892	−	0.933763i	$-0.383496\pi$
$991$	22.5330	0.715784	0.357892	+	0.933763i	$0.383496\pi$
$992$	5.65685	0.179605
$993$	69.2368	2.19716
$994$	0	0
$995$	0	0
$996$	−25.2164	−0.799011
$997$	0.589552	0.0186713	0.00933565	−	0.999956i	$-0.497028\pi$
$997$	0.589552	0.0186713	0.00933565	+	0.999956i	$0.497028\pi$
$998$	13.2665	0.419944
$999$	46.9042	1.48398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bt.1.4	yes	4
5.2	odd	4		2450.2.c.x.99.1		8
5.3	odd	4		2450.2.c.x.99.8		8
5.4	even	2		2450.2.a.bu.1.1	yes	4
7.6	odd	2	inner	2450.2.a.bt.1.1	✓	4
35.13	even	4		2450.2.c.x.99.5		8
35.27	even	4		2450.2.c.x.99.4		8
35.34	odd	2		2450.2.a.bu.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2450.2.a.bt.1.1	✓	4	7.6	odd	2	inner
2450.2.a.bt.1.4	yes	4	1.1	even	1	trivial
2450.2.a.bu.1.1	yes	4	5.4	even	2
2450.2.a.bu.1.4	yes	4	35.34	odd	2
2450.2.c.x.99.1		8	5.2	odd	4
2450.2.c.x.99.4		8	35.27	even	4
2450.2.c.x.99.5		8	35.13	even	4
2450.2.c.x.99.8		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.05231 q^{3} +1.00000 q^{4} -3.05231 q^{6} -1.00000 q^{8} +6.31662 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.05231 q^{3} +1.00000 q^{4} -3.05231 q^{6} -1.00000 q^{8} +6.31662 q^{9} +5.31662 q^{11} +3.05231 q^{12} +3.27620 q^{13} +1.00000 q^{16} -7.29496 q^{17} -6.31662 q^{18} +1.63810 q^{19} -5.31662 q^{22} +8.63325 q^{23} -3.05231 q^{24} -3.27620 q^{26} +10.1234 q^{27} -2.63325 q^{29} -5.65685 q^{31} -1.00000 q^{32} +16.2280 q^{33} +7.29496 q^{34} +6.31662 q^{36} +4.63325 q^{37} -1.63810 q^{38} +10.0000 q^{39} +2.60454 q^{41} -6.63325 q^{43} +5.31662 q^{44} -8.63325 q^{46} -6.10463 q^{47} +3.05231 q^{48} -22.2665 q^{51} +3.27620 q^{52} -8.00000 q^{53} -10.1234 q^{54} +5.00000 q^{57} +2.63325 q^{58} +1.86199 q^{59} +6.10463 q^{61} +5.65685 q^{62} +1.00000 q^{64} -16.2280 q^{66} -1.63325 q^{67} -7.29496 q^{68} +26.3514 q^{69} -2.00000 q^{71} -6.31662 q^{72} -7.29496 q^{73} -4.63325 q^{74} +1.63810 q^{76} -10.0000 q^{78} +1.36675 q^{79} +11.9499 q^{81} -2.60454 q^{82} -8.26139 q^{83} +6.63325 q^{86} -8.03751 q^{87} -5.31662 q^{88} +4.91430 q^{89} +8.63325 q^{92} -17.2665 q^{93} +6.10463 q^{94} -3.05231 q^{96} +4.24264 q^{97} +33.5831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 12 q^{9} + 8 q^{11} + 4 q^{16} - 12 q^{18} - 8 q^{22} + 8 q^{23} + 16 q^{29} - 4 q^{32} + 12 q^{36} - 8 q^{37} + 40 q^{39} + 8 q^{44} - 8 q^{46} - 36 q^{51} - 32 q^{53} + 20 q^{57} - 16 q^{58} + 4 q^{64} + 20 q^{67} - 8 q^{71} - 12 q^{72} + 8 q^{74} - 40 q^{78} + 32 q^{79} + 8 q^{81} - 8 q^{88} + 8 q^{92} - 16 q^{93} + 68 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 12 * q^9 + 8 * q^11 + 4 * q^16 - 12 * q^18 - 8 * q^22 + 8 * q^23 + 16 * q^29 - 4 * q^32 + 12 * q^36 - 8 * q^37 + 40 * q^39 + 8 * q^44 - 8 * q^46 - 36 * q^51 - 32 * q^53 + 20 * q^57 - 16 * q^58 + 4 * q^64 + 20 * q^67 - 8 * q^71 - 12 * q^72 + 8 * q^74 - 40 * q^78 + 32 * q^79 + 8 * q^81 - 8 * q^88 + 8 * q^92 - 16 * q^93 + 68 * q^99

Introduction

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